What’s the use of chaos?

Chris Budd

Is life predictable or unpredictable?

Can we tell what is going to happen

• In the next second?

• In the next hour?

• In the next year?

Does nature have an underlying order and pattern?

First answer …. YES!

If we look we can see order and pattern all around us …..

Science is the search for order and pattern in the universe

Snow crystals

The animal world

The motion of the planets

Galileo Pisa

One of the first to realise this

1600

Galileo watched a pendulum swing and realised that it was governed by predictable laws

Swing time of the pendulum was constant

• Regardless of how it was pushed

• Or where it was

• Or when

Newton

1686

In the Principia Newton showed that this order and pattern could be expressed by using mathematics

0)sin(2

2

g

dt

dk

dt

dl

Pendulum equation

Key idea ….

• Write down the equations describing a physical system

• Solve the equations

• Predict the future

Does this work?

Neptune: discovered by maths

32

2

x

GMx

dt

xd Newton’s law of gravitation

Weather forecasting

0.,Re

1. 2 uuPuuut

Navier-Stokes equations

Laplace’s prediction …

If we were to know the exact position of every particle in the universe then we could predict the future with certainty

No room here for free will!

Lots of natural and human events seem to be very unpredictable!!

Do you feel lucky?

Clouds and the weather (after a week)

El Nino

Ocean temperature

Year

Climate change

The stock market

My dog

Does this complex behaviour arise because nature is really complicated and unexplainable

does it arise naturally from Newton’s laws???

Or …….

The Double Pendulum

Motion can be

• Periodic in phase : predictable

• Periodic out of phase : predictable

• Chaotic : unpredictable

Newton’s laws apply to the double pendulum!

0)sin()sin()cos( 112

2

2122

22

21

2

dt

dm

dt

dm

dt

d

0)sin()sin()cos( 212

2

1122

12

22

2

dt

d

dt

d

dt

d

21 Angle of top part

Angle of bottom part

Chaos

Simple rules can lead to complex and unpredictable behaviour

Poincare: discoverer of chaos

The problems of being a town planner

nx

nx

Population of the town in the year n

Can we relate this years population:

To next years population: 1nx

Can we predict the population of a town?

Year

Population

Malthus ….

nn axx 1

• a = 1 … population stays constant

• a > 1 … population increases

• a < 1 … population decreases

Birthrate/Deathrate

)(1 nnn xMaxx

Problem … population runs out of resources

Improved model proposed by May

Maximum population

What does this predict?

a = 2 Single point

a = 3 Two points

a = 3.55 8 points

a = 3.8 Chaos

So, what’s the use of chaos?

Chaos: Simple rules can give complex patterns

Barnsley fractal fern

Mandlebrot set Fractal mountain

Computer graphics

Turbulence: da Vinci

New understanding of nature

River delta

Motion of the asteroids

Chaos is the science of the 21st Century!